Deteriorating hospitals at home drive pregnant women across border, where they are straining Colombia’s public-health network

CÚCUTA, Colombia—Fearful of having a baby in a grim Venezuelan hospital, Gerlimar Pastrán took a nine-hour bus ride to this border city in Colombia to give birth earlier this month to a healthy boy in a functioning maternity ward.

She is one of hundreds of pregnant Venezuelan women who have streamed into Colombia in recent months to have...